The Range of Topological Effects on Communication

We continue the study of communication cost of computing functions when inputs are distributed amongk processors, each of which is located at one vertex of a network/graph called a terminal. Every other node ofthe network also has a processor, with no input. The communication is point-to-point and the cost is the totalnumber of bits exchanged by the protocol, in the worst case, on all edges.Chattopadhyay, Radhakrishnan and Rudra (FOCS’14) recently initiated a study of the effect of topology ofthe network on the total communication cost using tools fromL1 embeddings. Their techniques provided tightbounds for simple functions like Element-Distinctness (ED), which depend on the 1-median of the graph. Thiswork addresses two other kinds of natural functions. We show that for a large class of natural functions like Set-Disjointness the communication cost is essentially n times the cost of the optimal Steiner tree connecting theterminals. Further, we show for natural composed functions like ED ◦XOR and XOR ◦ED, the naive protocolssuggested by their definition is optimal for general networks. Interestingly, the bounds for these functionsdependonmore involved topological parameters that are a combination of Steiner tree and 1-median costs.To obtain our results, we use somenew tools in addition to ones used in Chattopadhyay et. al. These include (i)viewing the communication constraints via a linear program;(ii) using tools from the theory of tree embeddings toprove topology sensitive direct sum results that handle the case of composed functions and (iii) representing thecommunication constraints of certain problems as a family of collection of multiway cuts, where each multiwaycut simulates the hardness of computing the function on the star topology. ∗School of Technology and Computer Science, Tata Institute of Fundamental Research, email: [email protected]. Research partially sup-ported by a Ramanujan Fellowship of the DST.†Department of Computer Science and Engineering, University at Buffalo, SUNY, email: [email protected]. Research supported in part byNSF grant CCF-0844796.